1360 BOOK REVIEWS BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 84, Number 6, November 1978 © American Mathematical Society 1978 Catastrophe theory: Selected papers, 1972-1977, by E. C. Zeeman, Addison- Wesley, London, Amsterdam, Ontario, Sydney, Tokyo, 1977, x + 675 pp., $26.50 (hard binding), $14.50 (paper binding). For the general public, catastrophe theory (or CT) has become the biggest thing in mathematics. René Thorn and Christopher Zeeman are the two leaders of this field. L'Express (October 14-30, 1974) asserts that the "new Newton" is French (i.e. Thorn). An announcement of Zeeman's lecture at Northwestern University in the spring of 1977 contains a quote describing catastrophe theory as the most important development in mathematics since the invention of calculus 300 years ago. Newsweek has given similar comparisons. Zeeman juxtaposes Newton and Thorn in the volume under review (briefly ZCT), p. 623. Thorn writes " ... CT is-quite likely-the first coherent attempt (since Aristotelian Logic) to give a theory on analogy" [p. 637, ZCT]. On the back cover of Thorn's book, Structural stability and morphogenesis [English translation, Benjamin, 1975 or Thorn's SSM], is the quote from the London Times review, "In one sense the only book with which it can be compared is Newton's Principia" Recently however, the importance of CT has been sharply and publicly challenged by Hector Sussman and subsequently by Sussman and Raphael Zahler [Catastrophe theory as applied to the social and biological sciences: A critique to appear in Synthese]. A critical story on CT by Gina Kolata in "Science", April 15, 1977, is headed: Catastrophe theory: The emperor has no clothes. A front page story on the New York Times, November 19, 1977 focuses on the challenges to CT. To write a review in this environment has a very personal side for me. On one hand my own work on dynamical systems is closely connected to the origins of CT. I have had a long and close personal and professional relationship with both Thorn and Zeeman. More than 20 years ago I was discussing singularities of maps, transversality, and immersions with René Thorn. Thorn tried to interest me in an early draft of chapters of his book Structural stability and morphogenesis in 1966. On the other hand I have remained skeptical and aloof from CT, perhaps due to my conservatism in science. While my colleagues and students were showing enthusiasm for CT, I gave critical lectures, one at the University of Chicago in 1974, one at the Aspen Institute of Physics in 1975. More recently I have been quoted negatively in the "Science" and New York Times references above. This is the first time I have written on the subject, and I should warn the reader of this negative bias, far from shared by many of my fellow mathematicians. Some of the mathematics underlying CT, especially transversality, and singularities of maps, has played a constructive role in outside disciplines, and is destined to play an ever increasing role. On the other hand I feel that CT itself has limited substance, great pretension and that catastrophe theorists have created a false picture in the mathematical community and the public as License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use BOOK REVIEWS 1361 to the power of CT to solve problems in the social and natural sciences. It is to the credit of Sussman and Zahler that they have seriously challenged this false picture. On this matter of warning to the reader, there is also the problem of my "quoting out of context". In fact the quotes I have made above and will make below are often imbedded in more prosaic language, perhaps tempering the brief and sometimes sharp sounding statements I refer to. At this point, I would like to acknowledge my debt to J. Guckenheimer, C. Zeeman, H. Sussman, and M. Hirsch among others for very useful conversations on the subject here. Besides the work of Thorn and Zeeman, on Catastrophe Theory, I have also learned much from that of Sussman and Zahler. Just what is Catastrophe Theory? Thorn writes that CT "... has to be considered as a theory of general morphology" 633, ZCT and Zeeman "CT is a new mathematical method for describing the evolution of forms in nature." (1, ZCT). However the new mathematics associated to CT really is contained in what is called elementary catastrophe theory (ECT) by Thorn and Zeeman. Very briefly, ECT studies a smooth real valued map ƒ as a function (often called a "potential function") of a state x and parameter JU,. Here the state lies in some Euclidean space and jut also varies in a (usually low dimensional) Euclidean space. The problem is to find local canonical forms for "generic" ƒ, using a smooth change of coordinates in the variables x, /x, y = ƒ (x, /x). In this case, "generic" means for an open dense subset in a suitable function space. A local canonical form is defined in a neighborhood of a pair (x0, /XQ) at which ƒ is singular. ECT solves this problem when /i is in a low ( < 5) dimensional space. In particular, when /i lies in a 4 dimensional space (e.g. space-time), Thorn found seven canonical forms. We will shortly describe the case of the cusp catastrophe where x is in R and /x in R2. Is CT much more than ECT? Here Thorn and Zeeman differ. Thorn writes "... that Christopher's criticisms arise basically from a strict dogmatic view of CT which he identifies with ECT..." 633, ZCT. Zeeman replies (same page) that his emphasis on ECT has been mainly because of its usefulness in applications. In fact, the book under review, although titled CT, deals almost completely with ECT (its mathematics and applications). In my mind, when CT goes beyond ECT it loses pretty much any direct touch with mathematics. It is true that Thorn refers to nonelementary CT as "generalized catastrophes, composed map catastrophes, (7-invariant catastrophes" etc. 633, ZCT. But here the relationship of the mathematics to nature in the CT literature becomes tenuous and infrequent. For example in Thorn (SSM), the words "generalized catastrophe" are used to describe situations where no mathematical model is proposed. More particularly under the picture of "feather buds on a chicken embryo", Thorn writes "a generali­ zed catastrophe and example of symmetry in biology" (Figure 21, Thom's SSM). He labels a picture of "the Crab nebula, the remains of the explosion of a supernova" with "a partially filament catastrophe in astrophysics" (Figure 17, Thom's SSM). Thorn described the filament catastrophe as a License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1362 BOOK REVIEWS certain "codimension two" type of generalized catastrophe, but the mathematics is not specified. No doubt it is in this spirit Thorn writes (p. 189 in Structural stability, catastrophe theory and applied mathematics, SIAM Rev., April, 1977) "The truth is that CT is not a mathematical theory, but a body of ideas, I dare say a state of mind." In ECT, one can take the gradient of the "potential" function with respect to the state variable to obtain a dynamical system JC' = grad f^{x) parame­ terized by jm fp(x) » /(JC, (i). It is in this context that Thorn introduced ECT in his book SSM, Chapter 5. He takes /i to be in R4 or space-time. Elementary catastrophes are defined there as the points JLL of space-time for which the qualitative dynamics of JC' = gradj^(jc) changes in a neighborhood of the attractors (stable equilibria) of that dynamics. Generalized catastrophes are defined analagously where the gradient dynamical systems are replaced by a more general class of dynamical systems. However the mathematical theorems that Thorn states (later proved by Mather) actually give the classification of elementary catastrophes according to the theory of singularities of maps in contrast to the theory of dynamical systems. Guckenheimer subsequently pointed out (Bull. Amer. Math. Soc. 79 (1973), 878-890, and in Bifurcation and catastrophe, Dynamical Systems (éd., Peixoto) Academic Press, 1973) that the two classifications differ already when /i is in R3 (and certainly for /i in R4). I believe this weakens much of the scientific or philosophical foundations of CT which is a theory based on dynamical systems (Thorn's SSM). If even the elementary catastrophes don't correspond to the dynamical classification, what of the generalized catastrophes where the mathematics in SSM deals in only a few known examples? In fact ECT seems much closer to what economists call comparative statics than to dynamical systems. One can sense a corresponding change of perspective in the work of the catastrophe theorists. As we have noted, the starting point of CT in Thorn's SSM was the "bifurcations" of dynamical systems parameterized by space- time. In Zeeman's work, these parameters changed from "space-time" to "control parameters". Now in his SIAM article (cited above) Thorn also speaks of control parameters and seems to de-emphasize the dynamical system foundations of CT, especially relative to space-time. The cusp catastrophe is the most important example of a catastrophe and much if not most of the book under review (ZCT) centers around the cusp catastrophe and its applications. Zeeman with Isnard (ZCT, 329 in Some models from catastrophe theory in the social sciences) describes the canonical model as follows. In 3 dimensional space R3, variables (a, b, JC), let M be the cubic surface defined by JC3 = a + bx. Here a, b are horizontal axes and are the controls; x is the vertical state variable.